# Tagged Questions

This tag is for questions about the foundations of mathematics, and the formalization of mathematical concepts in foundational theories (e.g. set theory, category theory, and type theory).

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### Switch algebraic sign

I can't believe that I seriously ask this question as it is so simple. Given this $-x^3+4x$ I'd like to factor out -x, so I did $-x(x^2-4)$ which equals $-x(x^2-2^2)$ equals $-x(x-2)(x+2)$ ...
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### Is there a schsim between classical logic and categorical logic?

I've been trying to learn a little bit more about the foundations of mathematics, and it has strike me that there seems to be two competing points of view about what the foundations should be. While ...
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### Preserving equality between different mathematical objects

I'm taking an 'Intro to Higher Mathematics'-type course right now, were we learn about basic set theory, number theory, algebra, etc. and I had the following thought: Say you're trying to solve a ...
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### Could mathematical reasoning be non-axiomatic?

"Mathematics is not a deductive science—that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, ...
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### Existential axioms for category theory

There are some existential axioms in set theory, for example, axiom schema of specification. It's my understanding that category theory isn't based essentially on set theoretic foundation. If so, I ...
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### Is there a model of ZFC inside which ZFC does not have a model?

Assuming ZFC has a model, is there a model of ZFC such that in that model, ZFC has no model? Also, assuming ZFC has a model, is there a model of ZFC such that in that model, ZFC is inconsistent?
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### The monadic second order theory with $<$ and Presburger arithmetic

Consider the monadic second order logic over the natural numbers with $<$ as a predicate, i.e. the second order logic over $(\mathbb N, 1, <)$, where we can quantify over sets and individual ...
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### Any foundational theory of math falls prey to the incompleteness theorems - true or false?

I heard somewhere on the internet once something along the following lines: Any conceivable foundational theory of mathematics (be it ZFC or, if ZFC was found to be inconsistent, some modification ...
By definition, an isomorphism between two objects in a category is a morphism so and so.. We know that $\mathbf {Cat}$ is the categories of small categories so that morphisms between objects are ...